Crash Course in Kinematics Johar Ashfaque Kinematics lies at the heart of the modern S-matrix program. A staple of every introductory course in QFT, this subject is typically given short shrift in the mad dash to dynamics. However, as we will see, any sharp distinction between kinematics and dynamics is very much artificial. Many scattering amplitudes are actually uniquely fixed by kinematic constraints like Lorentz invariance and momentum conservation. In the textbook formulation of QFT, kinematic data is characterized more or less by momentum vectors pµ = (p0 , p~) where throughout we will take the convention that all particles are incoming. Over the years, the study of scattering amplitudes program has revealed a menagerie of alternative variables like spinor helicity and momentum twistors to a name a few. Of course, these all describe the same underlying physics. However, by recasting expressions into the appropriate variables, one can achieve massive simplifications which reveal otherwise invisible structures.
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Spinor Helicity
Spinor helicity variables are a tremendously powerful tool for representing on-shell kinematics. We will focus here on the four-dimensional spinor helicity formalism but generalizations exist for lower and higher dimensions. In a nutshell, the spinor helicity formalism maps the the components of a four-vector into those of a two-by-two matrix via p0 + p3 p1 − ip2 µ pαα̇ = pµ σαα̇ == p1 + ip2 p0 − p3 where σ µ = (1, ~σ ) is a four-vector of Pauli matrices and the undotted and dotted indices transform under the usual spinor representations of the Lorentz group. The only Lorentz invariant quantity which can be constructed from pαα̇ is its determinant det(p) = pµ pµ = 0 which vanishes for a massless on-shell particle. The spinor helicity formalism has been generalized to include massive particles. However, the honest truth is that theories with massless particles are unambiguously simpler than their massive counterparts. This fact transcends the context of scattering amplitudes. Even in the action formulation, particles like gluons, gravitons, chiral fermions, or Nambu-Goldstone bosons are all massless as a consequence of symmetries which also stringently constrain their permitted interactions. This is the reason why we so often define massive particles as a controlled perturbation away from the massless limit take for example the case of the soft breaking of supersymmetry or chiral symmetry. Returning to det(p) = pµ pµ = 0 we that pαα̇ has a vanishing determinant. Since pαα̇ is non-zero it is a two-by-two matrix of at most rank one which without loss of generality can be written as the outer product of two two-component objects which are called spinors pαα̇ = λα λ̃α̇ . Note that λα and λ̃α̇ are sometimes called holomorphic and anti-holomorphic spinors respectively due to their transformation properties under the Lorentz group. Bear in mind that there is no connection here to any underlying fermionic states - these spinors are not anti-commuting Grassmann numbers but are just complex numbers. For real momenta pαα̇ is Hermitian implying the reality condition λ̃α̇ = ±(λ∗ )α̇ 1
for positive and negative energy states respectively. When the momentum is complex there is no relation between Νι and ÎťĚƒÎąĚ‡ and they are independent. However, in this case pιι̇ still satisfies det(p) = pÂľ pÂľ = 0 so these spinors still describe a massless particle. Strictly speaking, since the external momenta in physical processes are real, we should maintain this reality condition throughout the calculations. However, much of the power will flow from analytic continuation to complex momenta while treating the physical S-matrix as a restriction of this more general object. Let us now introduce the Lorentz invariant building blocks of the spinor helicity formalism. Given two particles i and j, define hiji = ÎťiÎą Îťjβ ιβ [ij] = ÎťĚƒiι̇ ÎťĚƒiβ̇ ι̇β̇ which are commonly referred to as the “angleâ€? and “squareâ€? brackets respectively. Any function of fourdimensional kinematic data can be written exclusively in terms of these objects. For example, angle and square brackets come together to form the familiar Mandelstam invariants sij = (pi + pj )2 = 2pi pj = hiji[ij] and likewise for higher-point kinematic invariants. Because spinors are simple objects, they are subject to relatively few algebraic manipulations. One important property is antisymmetry: hiji = −hjii and [ij] = −[ji] which in turn implies that hiii = [ii] = 0. Moreover since each spinor is two-dimensional, one can always write a spinor as a linear combination of two linearly-independent spinors hijiÎťk + hkiiÎťj + hjkiÎťi = 0 which is known as the Schouten identity. In fact, spinor helicity variables are nothing more than an algebraic reshuffling of the external kinematic data. Such a manipulation would not be particularly advantageous were it not for the fact that scattering amplitudes enjoy an immense reduction in complexity when translated into these variables. The avatar of this simplification is the S-matrix for gluon scattering. Parke and Taylor were first to realize that for maximally helicity violating (MHV) configuration that is two negative helicity gluons with the rest positive, the mountain of algebra telescopes into an expression built from simple color structures multiplying monomial expressions of the form A(¡ ¡ ¡i− ¡ ¡ ¡ j − ¡ ¡¡) =
hiji4 h12ih23i ¡ ¡ ¡ hn1i
which is the celebrated Parke-Taylor formula. Notice first that almost all the terms in the Feynman diagram expansion are nothing more than a complicated rewriting of the zero. Secondly, new features of the amplitude become manifest: in particular, the Parke-Taylor formula is a function of angle brackets revealing an underlying holomorphic structure of MHV amplitudes.
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The Little Group
Spinor helicity variables have a certain aesthetic elegance. But why are they practically useful? As is so often the case, their utility flows from the fact that they linearly realize the symmetries of the system. These symmetries are Lorentz invariance and the so-called little group. The little group is comprised of the subset of Lorentz transformations which leave the momentum pÂľ of a particle invariant. The transformations which leave pÂľ invariant form the ISO(2) group of translations and rotations in the plane transverse to the trajectory of the particle. The finite dimensional representations of this group are eigenvectors of the rotation subgroup. The corresponding eigenvalues label the helicities of physical states. Under the little group, the spinor helicity variables Îťi and Îťi transform so as to leave pi is unchanged. This is not very useful if interacting scalars are considered as these particles are singlets of the little group and the amplitude is solely a function of pi . On the other hand, for particles with spin 2
it is necessary to introduce non-trivial representations of the little group which carry helicity quantum numbers. In terms of conventional Feynman diagrams this little group covariance enters through an additional set of kinematic objects known as the polarizations. In terms of spinor helicity variables, polarization vectors take the form e+ αα̇ =
ηα λ̃α̇ hηλi
e− αα̇ =
λα η̃α̇ [λ̃η̃]
where the ± superscripts label the helicity. By construction, these polarizations are transverse to the momenta, so pαα̇ e+ αα̇ ∝ [λ̃λ̃] = 0, pαα̇ e− αα̇ ∝ hλλi = 0. Note the appearance of reference spinors, η and η̃ which are linearly independent of λ and λ̃ but otherwise arbitrary. In fact, one can even assign different reference spinors for each external particle. Reference spinors can be tremendously useful for explicit calculations.
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